Structural Response to Vapour Cloud Explosions · 2019. 11. 12. · Structural Response to Vapour...

Structural Response to

Vapour Cloud Explosions

Anqi Chen

Department of Civil and Environmental Engineering

Imperial College London

A thesis submitted for the degree of

Doctor of Philosophy

2014

Abstract

Over the last few decades, a number of major industrial blast accidents

involving oil and gas installations occurred worldwide. These include

the blast explosion that occurred in an industrial facility at Buncefield

in the United Kingdom in 2005. Extensive damage occurred due to

the blast, both to the industrial plant and surrounding buildings, as

a consequence of much higher overpressures than would normally be

expected from a vapour cloud explosion of this nature. In response

to this event, a great deal of work was carried out on collecting and

analysing available evidence from the incident in order to understand

the explosion mechanism and estimate the overpressure levels within

the gas cloud that formed. Subsequent investigations included the

examination of steel switch boxes on the site located within the area

covered by the vapour cloud. These boxes suffered varying degrees

of damage and could therefore be used as overpressure indicators. A

series of tests were commissioned after the event in order to compare

the damage of the field boxes with detonation tests on similar boxes.

The thesis firstly reports on numerical studies carried out on assess-

ing the damage to steel boxes subjected to both detonation and de-

flagration scenarios in order to aid the investigation of the explo-

sion. Several modelling approaches are adopted in the numerical stud-

ies, including: Pure Lagrangian, Uncoupled Lagrangian-Eulerian and

Coupled Lagrangian-Eulerian techniques. The numerical models are

validated against data collected from gas detonation experiments on

similar steel boxes. It is found that the coupled approach is able to

predict the results accurately, although such an approach cannot be

used in detailed parametric investigations due to its prohibitive com-

putational demand. The pure Lagrangian approach is therefore used

instead, but the overpressure range in the parametric assessments is

limited to 4 bar (side-on) as an adequate level of accuracy from this

modelling technique cannot be ensured beyond this range. The re-

sults are summarised in the form of pressure-impulse diagrams, and

typical residual shapes are selected with the aim of aiding forensic

investigations of future explosion incidents.

The investigation is extended thereafter to the response of a steel-

clad portal frame structures located outside the gas cloud and which

suffered varying degrees of damage. A typical warehouse building is

studied through a pure Lagrangian approach. A non-linear finite el-

ement model of a representative sub-structure of the warehouse wall

is validated against a full scale test carried out at Imperial College

London. A series of pressure-impulse diagrams of the sub-structure

is then constructed based on the results of parametric non-linear dy-

namic assessments using the developed numerical model under various

combinations of overpressures and impulses. A new failure criterion

based on the total failure of the self-tapping screws is proposed in

conjunction with pressure-impulse diagrams. This failure condition

provides a more direct assessment of the damage to the side walls of

the warehouse. The pressure-impulse diagrams can be used to assess

the response of a typical warehouse structure to blast loading, and

to provide some guidance on the safe siting in a hazardous environ-

ment around oil and storage sites. Simplified approaches based on

single degree of freedom representations are also employed, and their

results are compared with those from the detailed non-linear finite el-

ement models. The findings show that the simplified approaches offer

a reasonably reliable and practical tool for predicting the response of

the side rails. However, it is illustrated that such idealisations are

not suited for assessing the ultimate response of cladding panels, as

the side rail-cladding interactions cannot be captured by simplified

approaches and necessitate the deployment of detailed numerical pro-

cedures.

Declaration of Originality

I hereby declare that this thesis and the work reported herein was

composed by and originated entirely from me. Information derived

from the published and unpublished work of others has been

acknowledged in the text and references are given in the list of

sources.

Copyright Declaration

The copyright of this thesis rests with the author and is made

available under a Creative Commons Attribution Non-Commercial

No Derivatives licence. Researchers are free to copy, distribute or

transmit the thesis on the condition that they attribute it, that they

do not use it for commercial purposes and that they do not alter,

transform or build upon it. For any reuse or redistribution,

researchers must make clear to others the licence terms of this work.

I would like to dedicate this thesis to my loving parents Jun and

Zhengming, and my dear wife Lijia for their love and support.

Acknowledgements

I would like to express my deepest gratitude and appreciation to

my supervisors Dr Luke Louca and Prof Ahmed Elghazouli for their

continuous encouragement, involvement, fruitful advice and support

throughout my research steps. I greatly appreciate they provided me

with the opportunity of research and working with them in such an

excellent environment at Civil Engineering Department of Imperial

College London.

I would like to acknowledge the financial and mental support I received

from my parents and my wife without them this work would have

never been completed.

I would like to also acknowledge the technical support from Dr Gra-

ham Atkinson from Health and Safety Laboratory, Dr Bassam Burgan

from Steel Construction Institute and Mr Bob Simpson from Health

and Safety Executive.

Finally, the test data used to validate the model for the analysis of

instrument boxes was generated in a Joint Industry Project entitled

“Dispersion and Explosion Characteristics of Large Vapour Clouds”.

The project was managed by the Steel Construction Institute and

sponsored by:

• BP Exploration Operating Company Limited• Health and Safety Executive (UK)• Petrleo Brasileiro S.A. Petrobras• Shell Research Limited• Statoil Petroelum AS

• Total Refining and Chemicals• Le Ministre de Lcologie, du Dveloppement Durable, des Trans-ports et du Logement (France)

• National Institute for Public Health and The Environment (Nether-lands)

The author wishes to thank the sponsors for allowing this data to be

used in the present research.

Contents

Contents vii

List of Figures xiii

Nomenclature xxix

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Buncefield incident . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Motivation and objectives . . . . . . . . . . . . . . . . . . . . . . 4

2 Literature review 8

2.1 Vapour cloud explosion . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Explosion mechanisms . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Methods for explosion analysis . . . . . . . . . . . . . . . . 11

2.2 Review of past incidents . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Buncefield incident . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Other major accidents . . . . . . . . . . . . . . . . . . . . 18

2.3 Blast wave and loading . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Blast wave characteristics . . . . . . . . . . . . . . . . . . 20

2.3.1.1 Normal shock . . . . . . . . . . . . . . . . . . . . 20

2.3.1.2 Blast wave and scaling law . . . . . . . . . . . . . 26

2.3.2 Dynamic blast loads . . . . . . . . . . . . . . . . . . . . . 29

2.3.2.1 Front surface load . . . . . . . . . . . . . . . . . 30

2.3.2.2 Rear surface load . . . . . . . . . . . . . . . . . . 32

vii

CONTENTS

2.3.2.3 Top and side surface loads . . . . . . . . . . . . . 32

2.3.2.4 Loads on the frame . . . . . . . . . . . . . . . . . 32

2.3.2.5 Non-ideal blast wave . . . . . . . . . . . . . . . . 35

2.4 Structural response . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.1 Analysis methods . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.1.1 Single degree of freedom method . . . . . . . . . 36

2.4.1.2 Pressure - impulse diagram . . . . . . . . . . . . 43

2.4.1.3 Finite element method . . . . . . . . . . . . . . . 45

2.4.2 Material properties at high strain rate . . . . . . . . . . . 46

2.4.3 Modelling structural components . . . . . . . . . . . . . . 47

2.5 Design guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Numerical tools 54

3.1 Non-linear finite element method . . . . . . . . . . . . . . . . . . 54

3.2 Solution methods for non-linear FE problems . . . . . . . . . . . . 59

3.2.1 Newton - Raphson method . . . . . . . . . . . . . . . . . . 60

3.2.2 Arc length method . . . . . . . . . . . . . . . . . . . . . . 61

3.2.3 Explicit direct integration method . . . . . . . . . . . . . . 63

3.3 Eulerian analysis for blast wave . . . . . . . . . . . . . . . . . . . 64

3.4 Approaches for study of blast responses . . . . . . . . . . . . . . . 66

3.4.1 Lagrangian approach . . . . . . . . . . . . . . . . . . . . . 68

3.4.2 Uncoupled Eulerian - Lagrangian approach . . . . . . . . . 69

3.4.3 Coupled Eulerian - Lagrangian approach . . . . . . . . . . 69

3.5 Numerical model validations . . . . . . . . . . . . . . . . . . . . 69

3.5.1 One dimensional model . . . . . . . . . . . . . . . . . . . . 70

3.5.2 Blast wave propagation . . . . . . . . . . . . . . . . . . . . 72

3.5.3 Reflection of ideal blast wave . . . . . . . . . . . . . . . . 73

3.5.4 Square plate subjected to close HE explosion . . . . . . . . 76

3.6 Finite element analysis packages . . . . . . . . . . . . . . . . . . . 82

4 Blast response of field objects 85

4.1 Experimental observations . . . . . . . . . . . . . . . . . . . . . . 86

4.1.1 Steel switch boxes . . . . . . . . . . . . . . . . . . . . . . . 87

viii

CONTENTS

4.1.1.1 Box150 . . . . . . . . . . . . . . . . . . . . . . . 87

4.1.1.2 Box200 . . . . . . . . . . . . . . . . . . . . . . . 88

4.1.2 TNT detonation tests . . . . . . . . . . . . . . . . . . . . . 89

4.1.3 Gas detonation tests . . . . . . . . . . . . . . . . . . . . . 91

4.2 Numerical methodology . . . . . . . . . . . . . . . . . . . . . . . 98

4.2.1 Numerical model of boxes . . . . . . . . . . . . . . . . . . 98

4.2.1.1 Model of Box150 . . . . . . . . . . . . . . . . . . 98

4.2.1.2 Model of Box200 . . . . . . . . . . . . . . . . . . 101

4.2.2 Lagrangian analysis . . . . . . . . . . . . . . . . . . . . . . 101

4.2.2.1 Consideration of internal pressure . . . . . . . . . 105

4.2.3 Uncoupled Eulerian - Lagrangian analysis . . . . . . . . . 106

4.2.4 Coupled Eulerian - Lagrangian analysis . . . . . . . . . . . 110

4.2.5 Comparison of computational demands . . . . . . . . . . . 111

4.3 Comparative assessment . . . . . . . . . . . . . . . . . . . . . . . 112

4.3.1 Comparison of numerical results . . . . . . . . . . . . . . . 112

4.3.1.1 Blast wave and dynamic loading . . . . . . . . . 113

4.3.1.2 Box response to gas detonations . . . . . . . . . . 119

4.3.2 Model selection for parametric studies . . . . . . . . . . . 126

4.4 Parametric study and discussion . . . . . . . . . . . . . . . . . . . 135

4.4.1 Reflection of deflagration blast waves . . . . . . . . . . . . 135

4.4.2 Pressure - Impulse diagram and residual shapes . . . . . . 138

4.4.3 Comparison with recovered boxes . . . . . . . . . . . . . . 145

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5 Modelling of warehouse structures 150

5.1 Damages to Buncefield warehouses . . . . . . . . . . . . . . . . . 150

5.2 Typical portal frame configuration . . . . . . . . . . . . . . . . . . 153

5.3 Description of warehouse components . . . . . . . . . . . . . . . . 154

5.3.1 Portal frames . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.3.2 Gable wall frames . . . . . . . . . . . . . . . . . . . . . . . 155

5.3.3 Roof bracing . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.3.4 Side wall bracing . . . . . . . . . . . . . . . . . . . . . . . 156

5.3.5 Eaves beam . . . . . . . . . . . . . . . . . . . . . . . . . . 157

ix

CONTENTS

5.3.6 Purlin and side rail . . . . . . . . . . . . . . . . . . . . . . 157

5.3.7 Cladding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.3.8 Self-tapping screw fasteners . . . . . . . . . . . . . . . . . 158

5.3.9 M12 Bolt-Cleat connections . . . . . . . . . . . . . . . . . 159

5.4 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.4.1 Strength increase factor (SIF) . . . . . . . . . . . . . . . . 160

5.4.2 Dynamic increase factor (DIF) . . . . . . . . . . . . . . . . 161

5.5 Numerical models of warehouse . . . . . . . . . . . . . . . . . . . 162

5.5.1 Double span full model . . . . . . . . . . . . . . . . . . . . 162

5.5.2 Single span full model . . . . . . . . . . . . . . . . . . . . 164

5.5.3 Sub-assemblage model . . . . . . . . . . . . . . . . . . . . 164

5.5.4 Component model . . . . . . . . . . . . . . . . . . . . . . 164

5.5.5 Comparison of displacement histories . . . . . . . . . . . . 165

5.6 Validation of sub-assemblage model . . . . . . . . . . . . . . . . . 166

5.6.1 Tests on cladding-side rail assemblage . . . . . . . . . . . . 166

5.6.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . 169

5.7 Refined sub-assemblage model . . . . . . . . . . . . . . . . . . . . 173

5.7.1 Fastener model and validation . . . . . . . . . . . . . . . . 174

5.7.2 Coupled response of fasteners . . . . . . . . . . . . . . . . 179

5.7.3 M12 Bolt-Cleat Connection model . . . . . . . . . . . . . . 183

5.8 Complete warehouse sub-assemblage model . . . . . . . . . . . . . 187

6 Response of warehouse structures 190

6.1 Response measurement and damage levels . . . . . . . . . . . . . 190

6.2 Modal analysis and loading scenarios . . . . . . . . . . . . . . . . 194

6.3 Side rail deflection . . . . . . . . . . . . . . . . . . . . . . . . . . 196

6.4 Cladding and fasteners . . . . . . . . . . . . . . . . . . . . . . . . 198

6.5 Side rail axial response and connection behaviour . . . . . . . . . 212

6.6 Combined response . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6.7 Pressure - impulse diagrams . . . . . . . . . . . . . . . . . . . . . 217

6.7.1 Side rail transverse displacement . . . . . . . . . . . . . . 218

6.7.2 Side rail axial displacement . . . . . . . . . . . . . . . . . 219

6.7.3 Percentage of total connector failure . . . . . . . . . . . . 219

x

CONTENTS

6.7.4 Iso-curves for TNT explosive charges . . . . . . . . . . . . 221

6.7.5 Combined pressure - impulse diagrams . . . . . . . . . . . 222

6.8 Strain rate hardening of fasteners . . . . . . . . . . . . . . . . . . 224

6.8.1 Estimation of local strain rate hardening . . . . . . . . . . 224

6.8.2 Influence of strengthened fasteners . . . . . . . . . . . . . 231

7 Simplified approaches and design considerations 236

7.1 Single degree of freedom approach . . . . . . . . . . . . . . . . . . 236

7.1.1 Wall cladding . . . . . . . . . . . . . . . . . . . . . . . . . 237

7.1.2 Side rail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

7.1.3 Component interaction . . . . . . . . . . . . . . . . . . . . 241

7.1.4 Response of simplified model of side rail . . . . . . . . . . 242

7.1.5 Response of simplified model of cladding . . . . . . . . . . 245

7.1.6 Pressure - impulse diagrams of simplified models . . . . . . 248

7.2 Assessment of blast overpressures . . . . . . . . . . . . . . . . . . 249

7.2.1 Blast wave simulation on rigid warehouse . . . . . . . . . . 249

7.2.2 Discussion of blast loading results . . . . . . . . . . . . . . 254

7.3 Design considerations for steel-clad industrial buildings . . . . . . 263

7.3.1 Pressure - impulse diagrams . . . . . . . . . . . . . . . . . 263

7.3.2 Example of application of P-I diagrams . . . . . . . . . . . 264

7.3.3 SDOF and FE models . . . . . . . . . . . . . . . . . . . . 267

7.3.4 Side-rail connections . . . . . . . . . . . . . . . . . . . . . 268

7.3.5 Cladding connections (fasteners) . . . . . . . . . . . . . . . 268

7.3.6 Other considerations . . . . . . . . . . . . . . . . . . . . . 269

8 Conclusions and further work 270

8.1 Blast response of field objects . . . . . . . . . . . . . . . . . . . . 270

8.2 Blast responses of warehouse structures . . . . . . . . . . . . . . . 271

8.3 Assessment of blast loading . . . . . . . . . . . . . . . . . . . . . 273

8.4 Recommendations for further work . . . . . . . . . . . . . . . . . 274

Appendix A: Results of assessment of blast overpressure 276

xi

CONTENTS

Appendix B: Loading scenarios for parametric study of sub-assemblage

model 284

References 286

xii

List of Figures

1.1 The fire during Buncefield incident (SCI, 2009) . . . . . . . . . . 4

1.2 Damages due to explosion (SCI, 2009) . . . . . . . . . . . . . . . 5

2.1 Blast waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Blast waves parameters in TNO multi-energy method (Berg, 1985) 13

2.3 Buncefield site with location of buildings (Atkinson, 2011) . . . . 16

2.4 Buncefield Explosion Damages (SCI, 2009) . . . . . . . . . . . . . 17

2.5 Stationary normal shock in a moving stream . . . . . . . . . . . . 20

2.6 Explosive shock front moving in still air . . . . . . . . . . . . . . . 22

2.7 Normal reflected shock on a rigid surface . . . . . . . . . . . . . . 24

2.8 Reflected pressure coefficients vs. angle of incidence (β) (USDOD,

2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.9 Shape of blast waves . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Passage of a blast wave over a rigid object . . . . . . . . . . . . . 29

2.11 Blast curves (Kinney and Graham, 1985) . . . . . . . . . . . . . . 30

2.12 Blast load-time history on front surface (Kinney and Graham, 1985) 31

2.13 Blast load-time history other surface (Kinney and Graham, 1985) 33

2.14 Blast load-time history on frame (Kinney and Graham, 1985) . . 34

2.15 Typical equivalent single-degree-of-freedom system . . . . . . . . . 36

2.16 Resistance curve R(δ) for a typical SDOF system . . . . . . . . . 39

2.17 Response for all regimes . . . . . . . . . . . . . . . . . . . . . . . 42

2.18 Pressure - impulse diagrams . . . . . . . . . . . . . . . . . . . . . 44

2.19 Pressure - impulse diagrams comparison . . . . . . . . . . . . . . 45

2.20 Material response at high strain rate (USDOD, 2008) . . . . . . . 46

2.21 Strain rate hardening models . . . . . . . . . . . . . . . . . . . . . 48

xiii

LIST OF FIGURES

3.1 Newton Raphson Method . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Modified Riks method . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Lagrangian v.s. Eulerian deformation . . . . . . . . . . . . . . . . 64

3.4 Eulerian time incrementation (Spinelli and Vitali, 2009) . . . . . . 65

3.5 One dimensional Eulerian model . . . . . . . . . . . . . . . . . . . 70

3.6 Mesh study of 1D Eulerian model . . . . . . . . . . . . . . . . . . 71

3.7 Comparison for shock strength decay . . . . . . . . . . . . . . . . 72

3.8 Normal reflection in Eulerian model pin = 34.5kPa . . . . . . . . 74

3.9 Oblique reflection (pin = 34.5kPa) . . . . . . . . . . . . . . . . . . 75

3.10 Comparison for shock strength decay (USDOD, 2008) . . . . . . . 76

3.11 Boyd’s experiment configuration (Boyd, 2000) . . . . . . . . . . . 77

3.12 Numerical model in AUTODYN . . . . . . . . . . . . . . . . . . 78

3.13 Comparison of responses and the modified model . . . . . . . . . 80

4.1 Damaged steel switch box in the vapour cloud (SCI, 2009) . . . . 86

4.2 Photos of steel switch boxes . . . . . . . . . . . . . . . . . . . . . 87

4.3 Box150 (300×300×150mm) . . . . . . . . . . . . . . . . . . . . . 884.4 Box200 (300×300×200mm) . . . . . . . . . . . . . . . . . . . . . 894.5 Configuration of TNT detonation test (SCI, 2009) . . . . . . . . . 90

4.6 Residual deformations of ”Box150” exposed to TNT detonation,

incident overpressure from left to right: 100kPa, 200kPa, 480kPa

and 750kPa (SCI, 2009) . . . . . . . . . . . . . . . . . . . . . . . 90

4.7 Simplifield layout of gas detonation test . . . . . . . . . . . . . . . 92

4.8 Configuration of box support (Courtesy of Steel Construction In-

stitute) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.9 Test in progress (Allason, 2013) . . . . . . . . . . . . . . . . . . . 93

4.10 Pressure - time histories recorded by transducers . . . . . . . . . . 94

4.11 Before and after photographs of Box200-1,3,5 and 7 (Allason, 2013) 96

4.12 Digital scan of boxes . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.13 Deformation measurement of scanned box . . . . . . . . . . . . . 97

4.14 Box150 numerical model details . . . . . . . . . . . . . . . . . . . 99

4.15 Box150 model boundary conditions with mesh . . . . . . . . . . . 100

4.16 Box numerical mode mesh study . . . . . . . . . . . . . . . . . . . 100

xiv

LIST OF FIGURES

4.17 Box200 numerical model with mesh . . . . . . . . . . . . . . . . . 102

4.18 Box faces subjected to reflected and side-on pressure loading in

pure Lagrangian approach . . . . . . . . . . . . . . . . . . . . . . 103

4.19 Load application in Lagrangian analysis (Box200-7, Transducer B11)103

4.20 Modelling of internal pressure . . . . . . . . . . . . . . . . . . . . 105

4.21 Calibration of blast wave in Eulerian model . . . . . . . . . . . . 107

4.22 3D Eulerian model . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.23 A quarter of 3D Eulerian model for UEL analysis Step 1 . . . . . 109

4.24 Overpressure loading curves for UEL analysis . . . . . . . . . . . 110

4.25 CEL numerical model . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.26 Numerical blast wave and loading results for Box200-1 (Transducer

B16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113


B9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114


B10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115


B11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.30 Displacement - time histories of box front face centre . . . . . . . 119

4.31 Comparison of residual shapes of Box200-1 . . . . . . . . . . . . 123




4.35 Numerical results of overpressures for TNT explosions . . . . . . 128

4.36 Numerical results of displacements for boxes in TNT explosions . 129

4.37 Numerical displacements error compared with CEL model . . . . 132

4.38 Numerical residual shapes for Box150 in TNT explosions. Peak

side-on pressure: 1 bar, 2 bar and 3 bar . . . . . . . . . . . . . . . 133

4.39 Numerical residual shapes for Box150 in TNT explosions. Peak

side-on pressure: 4 bar, 4.8 bar and 7.5 bar . . . . . . . . . . . . 134

4.40 Numerical Eulerian model for reflection of deflagration . . . . . . 136

4.41 Reflection coefficient for deflagration(non-ideal blast wave) . . . . 137

4.42 Overpressure dynamic loadings for parametric study . . . . . . . . 138

xv

LIST OF FIGURES

4.43 Pressure - Impulse diagrams for Box150 (back support) . . . . . . 143

4.44 Pressure - Impulse diagrams for Box150 (bottom support) . . . . 144

4.45 Numerical residual shapes of Box150 (back support) . . . . . . . . 146

4.46 Numerical residual shapes of Box150 (bottom support) . . . . . . 147

4.47 Comparison of numerical residual shapes to damaged boxes from

Buncefield (SCI, 2009) . . . . . . . . . . . . . . . . . . . . . . . . 148

5.1 Damage to Warehouse 1 (Atkinson, 2011) . . . . . . . . . . . . . 151

5.2 Damage to Warehouse 2 (Atkinson, 2011) . . . . . . . . . . . . . 152

5.3 Typical warehouse configuration (Burgan and Moutaftsis, 2011) . 153

5.4 Portal frame layout (Burgan and Moutaftsis, 2011) . . . . . . . . 155

5.5 Gable wall frames (Burgan and Moutaftsis, 2011) . . . . . . . . . 155

5.6 Roof plan (Burgan and Moutaftsis, 2011) . . . . . . . . . . . . . . 156

5.7 Side wall frames (Burgan and Moutaftsis, 2011) . . . . . . . . . . 156

5.8 Cross-sectional details of eaves beam, purlin and side rail [units:mm]

(Hadley, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.9 Built-up cladding details (Burgan and Moutaftsis, 2011) . . . . . 158

5.10 Self-tapping screw (CA, 2008) . . . . . . . . . . . . . . . . . . . . 159

5.11 M12 Bolt-Cleat connection (Hadley, 2004) . . . . . . . . . . . . . 159

5.12 Numerical modelling levels . . . . . . . . . . . . . . . . . . . . . . 163

5.13 Comparison of displacement histories of side wall models . . . . . 165

5.14 Full scale test configuration of cladding-side rail assembly (Elgha-

zouli et al., 2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.15 Profile of cladding TR35/207 (thickness=1 mm) [unit:mm] . . . . 167

5.16 Material models for cladding and side rail . . . . . . . . . . . . . 168

5.17 Numerical model used for validation . . . . . . . . . . . . . . . . . 169

5.18 Mesh of test assembly numerical model . . . . . . . . . . . . . . . 170

5.19 Comparison of numerical results and experimental measurements 171

5.20 Deformed shapes of test assembly (Riks method) . . . . . . . . . 172

5.21 Comparison of pressure - central displacement responses . . . . . 173

5.22 Details of sub-assembly model representing the side wall of warehouse174

5.23 Self-tapping screw connector model . . . . . . . . . . . . . . . . . 175

xvi

LIST OF FIGURES

5.24 Lap joint pulling test using static test components (validation of

detailed fastener model) . . . . . . . . . . . . . . . . . . . . . . . 177

5.25 Shear response estimation of the fastener for connector model . . 178

5.26 Definition of the connector shear response for fasteners . . . . . . 182

5.27 Finite element model of M12 Bolt-Cleat connection . . . . . . . . 184

5.28 Response estimation of M12 Bolt - Cleat connection . . . . . . . . 185

5.29 Connector model for M12 Bolt-cleat connection . . . . . . . . . . 186

5.30 Sub-assemblage model of the wall structure in the warehouse building187

6.1 Details of sub-assemblage model . . . . . . . . . . . . . . . . . . . 191

6.2 Structural component mid-span deflection and end rotation . . . . 192

6.3 Sub-assemblage mode shapes . . . . . . . . . . . . . . . . . . . . . 194

6.4 Loading scenarios: combinations of overpressure and impulse . . . 195

6.5 Displacement history at the mid-span of the central side rail with

loading profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.6 Deformed shapes of the sub-assemblage . . . . . . . . . . . . . . . 198

6.7 Response of central fastener of the sub-assemblage model in Case 98199

6.8 Response of central cladding of the sub-assemblage in Case 98 . . 201

6.9 Relative displacement at the middle of the cladding with failure of

the corresponding fastener . . . . . . . . . . . . . . . . . . . . . . 202

6.10 Response of central fastener of the sub-assemblage model in Case

157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

6.11 Response of central cladding of the sub-assemblage in Case 157 . 205

6.12 Response of side fastener of the sub-assemblage model in Case 75 206

6.13 Response of side cladding of the sub-assemblage in Case 75 . . . . 207

6.14 Comparison of failure behaviour of central and side cladding with

fasteners for the sub-assemblage model in Case 98 . . . . . . . . . 209

6.15 Cases with identical failure behaviour of cladding and fasteners at

centre and side . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

6.16 Response of middle fastener of the sub-assemblage model in Case 98211

6.17 Numerical response of the M12 bolt-cleat connection . . . . . . . 212

6.18 Axial and transverse displacement time history of side rail . . . . 214

6.19 Side rail transverse response with failure of fasteners . . . . . . . 216

xvii

LIST OF FIGURES

6.20 Percentage of total fastener failure with side rail displacement . . 217

6.21 Pressure - impulse diagrams: maximum side rail displacement . . 218

6.22 Pressure - impulse diagram: maximum axial displacement . . . . . 219

6.23 Pressure - impulse diagram . . . . . . . . . . . . . . . . . . . . . . 220

6.24 Iso-curves for TNT explosive charges . . . . . . . . . . . . . . . . 221

6.25 Combined pressure - impulse diagrams . . . . . . . . . . . . . . . 222

6.26 Strain rate effect on dynamic increase factor . . . . . . . . . . . . 224

6.27 Estimation of strain rate in the screw connection . . . . . . . . . 225

6.28 Stress, strain and strain rate time histories in the sampling shell

element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

6.29 Simplified model to estimate the strain rate in a screw connection 227

6.30 Applied 0.2 mm displacement over total time step of 0.01 second . 228

6.31 Strain rate analysis around the hole . . . . . . . . . . . . . . . . . 229

6.32 Relationship between nominal and concetrated strain rates . . . . 230

6.33 Connector response with and without the estimated strain rate effect230

6.34 Comparison of response of side rail with and without rate stiffened

fasteners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

6.35 Effect of the rate stiffened fastener on the pressure - impulse dia-

grams of the side rail maximum displacement . . . . . . . . . . . 233

6.36 Effect of the rate stiffened fastener on the percentage failure of

fasteners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

6.37 Influence of the rate effect on the pressure - impulse diagram of

maximum percentage of fastener failure (50% failure) . . . . . . . 234

7.1 Cladding panel and side rail simplified SDOF model . . . . . . . . 237

7.2 Resistance curve of SDOF model for cladding panel . . . . . . . . 239

7.3 Resistance curve of SDOF model for side rail . . . . . . . . . . . . 241

7.4 Response of simplified model of side rail . . . . . . . . . . . . . . 243

7.5 Comparison of pressure - impulse diagrams (maximum dynamic

displacement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

7.6 Responses of simplified model of cladding . . . . . . . . . . . . . . 246

7.7 Combined pressure - impulse diagram using the simplified approach248

xviii

LIST OF FIGURES

7.8 Numerical modelling of blast wave (semi-spherical 7500kg TNT

charge and 324m stand-off) and the rigid warehouse structure . . 251

7.9 Coupled model of blast wave with rigid warehouse . . . . . . . . . 252

7.10 Location of numerical pressure gauges . . . . . . . . . . . . . . . . 252

7.11 Pre-cut holes on the blastward wall of the rigid warehouse model . 253

7.12 30% venting area with different shapes of holes . . . . . . . . . . . 254

7.13 Simulation of blast loading on rigid warehouse model . . . . . . . 255

7.14 Overpressure loading results for Case 4 (30% venting area) . . . . 256

7.15 Internal pressurisation due to venting area on front wall . . . . . . 257

7.16 Overpressure loadings at the back wall . . . . . . . . . . . . . . . 259

7.17 Overpressure loadings of side wall . . . . . . . . . . . . . . . . . . 260

7.18 Effect of venting area shapes on internal pressurisation . . . . . . 262

7.19 Application of pressure - impulse diagram . . . . . . . . . . . . . 265

1 Overpressure loadings on rigid warehouse (Case 1) . . . . . . . . . 276






7 Overpressure loadings of back wall (Case 1 & 2) . . . . . . . . . . 279



10 Overpressure loadings of side wall (Case 1 & 2) . . . . . . . . . . 280



13 Overpressure loadings of front roof (Case 1 & 2) . . . . . . . . . . 281



16 Overpressure loadings of back roof (Case 1 & 2) . . . . . . . . . . 282



xix

LIST OF FIGURES

19 Combination of overpressures and impulses with corresponding

case number and positive duration . . . . . . . . . . . . . . . . . 285

xx

Nomenclature

Roman Symbols

[B] Strain matrix

[c] Constitutive matrix

[D] Assembled nodal displacement matrix

[De] Global nodal displacement matrix

[de] Local nodal displacement matrix

[F ] Assembled consistent nodal force vector

[fb] Body force vector

[Fe] Global consistent nodal force vector

[fe] Local consistent nodal force vector

[fs] Surface force vector

[I] Internal forces of the flexible body

[K] Assembled stiffness matrix

[Ke] Global stiffness matrix

[ke] Local stiffness matrix

[M ] Assembled mass matrix

xxi

LIST OF FIGURES

[T ] Transformation matrix

[U ] Displacement matrix

ūpl ABAQUS connector equivalent plastic relative motion

ΔPs Blast strength number (1-10)

ΔPs Maximum overpressure

Δt Step time increment

ΔP s Dimensionless maximum overpressure

u̇pl ABAQUS collection of plastic relative motion rate

R Energy scaled distance

t+ Dimensionless positive duration

A Cross sectional area

a, b TNO multi-energy correlation constants

a0 Local speed of sound

ax Local sound of speed (atmosphere)

C Strain rate hardening parameter for Johnson-Cook model

c0 Atmospheric speed of sound

Cd Dilatation wave speed

Cd Discharge coefficient

cp Specific heat at constant pressure

D Average obstacle size

D Diameter of the self-tapping screw shaft

D, p Strain rate hardening parameter for Cowper-Symonds model

xxii

LIST OF FIGURES

E Combustible energy of vapour cloud

E Detonation energy per initial volume

E Young’s modulus

F Axial pre-tension

F Distributed load

F Opening percentage

f ABAQUS collection of active connector forces

F 0 ABAQUS connector equivalent yield force

Fe Equivalent force of SDOF system

fe TNT efficiency factor

FN ABAQUS connector equivalent tensile (normal) force

FS ABAQUS connector equivalent shear force

fu Material ultimate stress

fx, fy, fz ABAQUS connector force

fy Material yield stress

fdu Material dynamic ultimate stress

fdy Material dynamic yield stress

g(t) Time function of the applied distributed load

HTNT Explosive heat of TNT

Hvapour Combustion heat of the fuel in vapour cloud

I Specific impulse

K Stiffness of SDOF model

xxiii

LIST OF FIGURES

K Torque coefficient

k Specific heat ratio (1.4 for ideal gas)

k Stiffness

K, r ABAQUS connector defining parameter

Ke Equivalent stiffness of SDOF system

KLM Load-mass factor

kTM Strain energy

L Characteristic length of the target

L Lagrange

L Length of a simply supported beam

Le Characteristic element length

Lf Flame path

m Mass

Me Equivalent mass of SDOF system

Me Global mass matrix

me Equivalent TNT mass

me Local mass matrix

Mp Ultimate plastic moment

Mx Upstream flow Mach number

mx,my,mz ABAQUS connector moment

My Downstream flow Mach number

mvapour Total mass of the vapour cloud

xxiv

LIST OF FIGURES

N Admissible functions

n nth incremental step

P ABAQUS connector potential

P Total applied load P=FL

p Blast wave overpressure

p(x, t) Applied distributed load

P0 Atmospheric pressure

Pa Atmospheric pressure

pA Atmospheric pressure

pi Incident overpressure

po Peak overpressure value

Pr Reflected pressure

pr Reflected overpressure

Px Upstream flow pressure

Py Downstream flow pressure

Patm Atmospheric pressure

Pincident Incident overpressure

Pstag Stagnation pressure

pstag Stagnation overpressure

Ptransmission Transmitted overpressure through perforated barriers

Qi Generalised force

qi Admissible functions

xxv

LIST OF FIGURES

R Ideal gas constant

R Stand-off distance

r Distance to target

R0 Charge radius

rc Radius of high explosive charge

Rm Maximum plastic load of SDOF resistance curve

RN ABAQUS connector equivalent tensile (normal) capacity

RS ABAQUS connector equivalent shear capacity

S Characteristic width of the reflected surface

Sc TNO multi-energy correlation scale factor

Sl Laminar burning velocity

sx Upstream flow entropy

sy Downstream flow entropy

T Kinetic energy

T Natural period

T Tension membrane force

T Tightening torque

t Time

t+ Positive duration

ta Shock front arrival time

td Shock wave positive duration

TH Temperature dependency parameter for Johnson-Cook material model

xxvi

LIST OF FIGURES

Tn Natural period of structure

tr Deflagration finite rise time

Tx Upstream flow temperature

Ty Downstream flow temperature

Tatm Atmospheric temperature

up Blast wave particle velocity

ux Upstream flow velocity (shock front velocity)

ux, uy, uz ABAQUS connector displacement

uy Downstream flow velocity

urx, ury, urz ABAQUS connector rotation

V Plastic dissipation energy

V Relative volume or the expansion of the explosive product

V Strain energy

W High explosive charge weight

W Work done by external forces

w(x, t) Response solution of a simply supported beam

Z Scaled distance

ūplDI ABAQUS connector damage initiation displacement

ūplf ABAQUS connector shear failure displacement

epeff Effective plastic strain

f failz ABAQUS connector tensile failure force

Greek Symbols

xxvii

LIST OF FIGURES

[�] Strain matrix

[σ] Stress matrix

α Blast wave form parameter

β Shock wave incident angle

δ Delta operator

δ(t) Generalised time response of a supported beam

δe Transverse displacement of beam at yieldings

δm Maximum transverse displacement of beam

δy Yielding displacement

δTM Strain energy

� The base of natural logarithm

�p Plastic strain

�y Material yield strain

Λn Pressure reflected ratio

μ Ductility ratio

ω Undamped natural circular frequency

Φ ABAQUS connector yield function

φ(x) Assumed mode shape function

π Potential energy

Ψm ABAQUS connector mode-mix ratio

ρ Density

ρx Density of the upstream flow

xxviii

LIST OF FIGURES

ρy Density of the downstream flow

ρatm Atmospheric density

σp Plastic stress

σy Material yield stress

Θ Temperature

θ Support rotation of beam

Θz Absolute zero

�̇ Strain rate

�̇Peff Effective plastic strain rate

�̇0 Reference strain rate

Acronyms

CEL Coupled Eulerian - Lagrangian

DIF Dynamic increase factor

NLFEM Non-linear finite element model

SDOF Single degree of freedom

UEL Uncoupled Eulerian - Lagrangian

V BR Volume blockage ratio

V CE Vapour cloud explosion

xxix

Chapter 1

Introduction

The 2005 Buncefield event was unexpectedly destructive for a vapour cloud ex-

plosion. It showed huge cost associated with blast damages to commercial and

residential buildings surrounding a major explosion incident. The main char-

acteristic of the Buncefield explosion was that its intensity and the severity of

damage caused to structures/buildings had not been anticipated by any major

hazardous design practice. The work presented in this thesis focuses on assessing

structural damage of vapour cloud explosions and aiming to provide insight into

the structure and objects response to blast loadings.

1.1 Background

An explosion is a sudden release of substantial energy, which transfers to its

surrounding in the form of a rapidly moving rise in pressure called a blast wave.

There are three categories of explosion: physical, nuclear and chemical (Smith

and Hetherington, 1994). The most common physical explosion can be found for

example in a failure of a vessel of compressed gas or other pressurized devices.

The violent expansion of the compressed gas is the source of the released energy.

The Nuclear explosion is much more destructive than the other two. The energy

in a nuclear explosion comes from the formation of different atomic nuclei by

redistribution of the neutrons and proton within the reacting nuclei.

Chemical explosions are exothermic reactions. The surrounding temperature

1

1. Introduction

is raised by the heat from oxidation of the fuel element contained within the

explosive compound. Then the volume of the gases becomes much larger than

the compound, generating a high pressure and subsequent blast wave. The energy

released arises from the formation of new molecular bonds in the products of the

reaction. Explosives that fall into this category are such as Trinitrotoluene (TNT,

high explosive), gunpowder (low explosive) and even the wheat flour dust in a

grain elevator. This explosion is known as a detonative process.

There is a special case of chemical explosion called vapour cloud explosion

(VCE) that involves a fuel (e.g. propane or butane) mixed with the atmosphere,

and usually this occurs in an offshore platform or industrial/chemical plant. If

the mixture is ignited, the flame velocity would be fast enough to generate a blast

wave. Although the overpressure in the blast wave might not be as high as in other

explosions, it can be strong enough to damage or even destroy nearby structures.

In a vapour cloud explosion, deflagration is the result that the flame front travels

slower than the speed of sound and if it moves faster than the speed of sound, we

would expect a detonation to occur (deflagration transition to detonation). It is

more destructive as it generates stronger blast waves than deflagration.

The blast wave (high overpressure) from an uncontrolled explosion can be

destructive and even lethal. To prevent its damage to buildings, there is a need

to understand the characteristics of explosions, blast waves and subsequently

the response of building structures. Empirical charts/tools are widely adopted

for analysis of characteristics of an explosion. The blast wave parameters can

be readily determined by using charts or empirical equations based on scaled

explosive charge weight and stand-off distance. Empirical methods are available

for both high explosive and vapour cloud explosions. With the development of

scientific computers, a more complex method known as Computational Fluid

Dynamics (CFD) is now widely available. It is capable of producing results with

better accuracy but involves greater cost of computational resource and operator

manpower.

Several methods are available for analysis of dynamic structural response.

Biggs’ Single Degree of Freedom (SDOF) model has been extensively used to

investigate the response of structural component to dynamic loadings. Results

can be presented in a graphical, analytical or numerical forms. This method is

2

1. Introduction

based on a representation of the actual component configurations by an equiv-

alent spring-mass system which has only one degree of freedom (Biggs, 1964).

An improved SDOF model is presented in the Technical Note 7 (SCI, 2002)

which overcomes some difficulties in Biggs’ model and includes complex support-

ing conditions. By using energy methods to generate equilibrium equations, the

mode approximation method has been extended by Schleyer (Schleyer and Hsu,

2000) and Langdon (Langdon and Schleyer, 2006b) and to incorporate full elas-

tic response and include variable support constraints. This method divides the

dynamic response of a component into different stages/modes and obtain results

by solving a Multi-Degree of Freedom system.

Finite element methods are also widely used to analyse the dynamic response

of structures. Commercial codes such as ?, ANSYS and LS-DYNA are commonly

used in research and industries. Borvik (Borvik et al., 2009) used LS-DYNA to

investigate whether a pure Lagrangian model can be used to determine the struc-

tural response in a specified blast problem or a more complicated fully coupled

Eulerian - Lagrangian model should be adopted. By using non-linear finite ele-

ment package Louca (Louca et al., 1996) studied the response of a typical blast

wall and a stiffened panel subject to hydrocarbon explosion. Langdon (Langdon

and Schleyer, 2006c) used ? to predict the response time histories and permanent

deformations of a stainless steel blast wall and connection systems.

1.2 Buncefield incident

Over the last few decades, there have been a number of major industrial accidents

involving oil and gas installations worldwide. These include the Buncefield event

in the UK in December 2005. The incident was a vapour cloud explosion which

occurred at the Buncefield Oil Storage Depot, Hemel Hempstead, Hertfordshire.

Fortunately there were no casualties but 43 people were injured. The explo-

sion and subsequent fire caused extensive damage to the plant and surrounding

buildings, both commercial and residential properties. An aerial view of the site

during the fire is shown in Figure 1.1. The economic impact of the incident

broadly adds up to £1 billion.

The subsequent investigation by the Buncefield Major Incident Investigation

3

1. Introduction

Figure 1.1: The fire during Buncefield incident (SCI, 2009)

Board (BMIIB) identified that the explosion was a consequence of oil spilling from

an over filled storage tank (tank 912) on the site. Both detonation and deflagra-

tion explosion mechanisms were assessed for their consistency with the observed

explosion characteristics. Neither of the two mechanisms can explain the evi-

dence conclusively. Work undertaken in the joint industry project (BMIIB, 2007)

has concluded that an early deflagration followed by transition to detonation at

congested area would be the most likely scenario.

1.3 Motivation and objectives

The work undertaken in the joint industry project (BMIIB, 2007) represented

a major advance in the understanding of the Buncefield explosion. The work

has also identified a number of areas that require further research. The work,

although defined within the frame of Buncefield, will lead to a greater under-

standing of large vapour cloud explosions and associated structural response.

Therefore, a further research project has been initiated by the recommendation

from the BMIIB. As part of the research work, interests of the present study are

4

1. Introduction

mainly lying in the structural response of various field objects and the steel portal

warehouse building.

The objective of the project is to study damage produced to buildings from

low-lying vapour cloud explosion which occurred in the previous accidents. Data

in the public domain on the damage observed will be used to investigate the

characteristics of the pressure time histories required to produce the damage

processes observed. Much effort will be placed in studying the impact of selected

pressure loads on representative building structures through detailed non-linear

finite element analysis. Analytical methods will be used in simple structures

where applicable. The anticipated outcome of this work would be to propose

guidance for the assessment of loading and the evaluation of structural response.

(a) Damaged Drum (b) Damaged Box (c) Damaged Warehouse

Figure 1.2: Damages due to explosion (SCI, 2009)

For the purpose of assessing the overpressure history across the incident site,

evidence relating to overpressure is important as part of the forensic studies.

Within the body of the vapour cloud, there were very few large objects such

as buildings. Buildings which suffered structural damage in the vicinity of the

site were located towards the edge or outside of the gas cloud. However, it was

found that there were many small objects such as steel drums and switch boxes

distributed across the site and nearby areas. These field objects could be used as

overpressure indicators, as their permanent deformation can provide an indication

of the possible overpressure at their locations after the passage of blast waves.

Typical damage of drums and boxes are shown in Figure 1.2a and 1.2b.

5

1. Introduction

During the explosion, both steel frame and masonry buildings were damaged

to different levels. Large steel portal frame warehouse type structures were af-

fected at a quite long distance. Among the damaged structures, steel frame

cladding warehouses are of the most interests. A damaged warehouse is shown

in Figure 1.2c. The high surface area and light-weight construction make the

pressure applied by wind a key structural design aspect. The common design

wind pressure of a steel cladding warehouse is lower than it is of a typical brick

building, and almost all of the surface structural steel and cladding is potentially

at risk (rather than just windows/doors). Therefore it is relatively more vulner-

able to accidental overpressure and the level of damage to a warehouse maybe be

particularly extensive and time consuming to repair, leading to longer periods of

business disruption.

Given the above considerations, field objects such as steel switch boxes were

chosen to perform a forensic study. While the steel frame cladding warehouse is

the subject of the analysis of structural response to vapour cloud explosions. The

main objectives of this project are:

• to model steel switch boxes subject to various blast loading

• to examine the blast response of steel boxes in the form of pressure-impulsediagrams and permanently deformed shapes

• to provide information for forensic studies to investigate the current andfuture explosions

• to model light weight steel portal frame warehouse structures under differentblast loadings

• to generate pressure-impulse diagrams of steel frame cladding warehouses

• to provide design implications for steel portal frame warehouses

In the following chapters, a literature review covering the recent research work

on study and investigation of blast explosion and structural response is presented,

following by a introduction of the numerical tools that adopted in the current

research project. Forensic works carried out on the field objects which can be

6

1. Introduction

used to help the investigation are presented. A detailed finite element analysis

on the response of light weight steel clad warehouse structure is presented and

the results are discussed. Some design implication and improvements to the

structures are included at the end of this thesis.

7

Chapter 2

Literature review

This chapter presents an overview of different types of explosions and their char-

acteristics. The Buncefield and other previous major incidents are reviewed and

their main characteristics are noted. Blast wave, the associated dynamic load-

ing and various tools for structural dynamic analysis are presented. A review

of previous work on studies of responses to blast loading is also included and

discussed.

2.1 Vapour cloud explosion

2.1.1 Explosion mechanisms

A cloud of combustible material (e.g. propane, butane and pentane etc.) in either

confined or unconfined atmosphere can explode under certain conditions. Con-

ditions leading to an unconfined vapour cloud explosion are first, a quick release

of a large quantity of fuel which forms a vapour cloud. Secondly the cloud is

ignited by a spark or flame. There are two mechanisms that a vapour cloud could

explode: deflagration and detonation. If the ignition occurs immediately after

the release of the combustible material before it mixes with the surrounding air,

the explosion is very likely to be a deflagration. When ignition of the combustible

cloud occurs after its formation so that the air has became well mixed with the

combustible vapour, and if the air-fuel mixture is just within the flammability

range, a detonation is most likely to occur.

8

2.Literature review

A vapour cloud explosion involving deflagration is certainly less destructive

than detonation. The overpressure is generated by the flame front travelling at

high speeds but under the local speed of sound. The mixture ahead is ignited by

heat radiation from the flame front. The flame generates pressure because of the

inertia of the unreacted mixture in front of the flame, similar to a object travelling

at high speed through the air generates a pressure wave in front of it. It is well

understood that deflagration of a vapour cloud in a confined or congested area

(such as a section fully packed with pipe works) can generate damaging pressures

due to acceleration of flame front. In some cases, the turbulence induced by

the congested obstacles will promote the reaction rate, producing successively

higher flame speeds and increasing overpressures. Thus it can eventually lead to

deflagration - detonation transition (DDT). It is important to note that the high

speed of flame is dependent on the congested obstacles, once the flame travels from

a confined area to an open area it rapidly decelerates. Hence the overpressures

decreases as the blast wave propagates away from the congested region.

As compared to deflagration, the detonation front has an initial sudden rise in

pressure, which then decays. This abrupt rise of overpressure is known as shock

front and the whole pressure wave as a shock wave. The shock front compresses

the air-fuel mixture in the front and raises the temperature. In detonation the

temperature rise can ignite the fresh mixture ahead as it exceeds the auto-ignition

temperature of the mixture. The magnitude of the shock front is maintained by

the energy released from the combustion process in detonation. The process is

self-sustainable so it is not dependent on any congestion or obstacles.

The pressure - time history of a typical blast wave from detonation as recorded

some distance away from the explosion centre is shown in Figure 2.1a. At an

arrival time ta after initiation of the explosion, the pressure at the remote location

suddenly increased to a peak value. The overpressure then immediately starts

to decay to atmospheric pressure over a period known as the duration of the

positive phase td. Due to the inertia of the air in the wave, a negative pressure

phase will occur after passage of the overpressure blast wave. The peak value

of the negative phase is much lower but the duration is much longer than the

positive phase. Generally three independent parameters are used to define a

detonation blast wave: the peak overpressure po, the positive phase duration td

9

2.Literature review

0.0 0.2 0.4 0.6 0.8 1.0

Time

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Overpressure

po

td

ta

+

-

Overpressure: po = Px − Py

Px

Py

AtmosphericPressure

(a) Detonation

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Overpressure

po

td

+

-

Overpressure: po = Px − Py

Px

Py

AtmosphericPressure

(b) Deflagration

Figure 2.1: Blast waves

10

2.Literature review

and the specific impulse which is the area under the positive phase I = 12potd.

The specific impulse is defined as the blast wave impulse per unit area.

In the case of deflagration the volume of the air-combustible mixture is gener-

ally high and the energy release is relatively low. This occurs because the reaction

rate in the deflagration is low compared with detonation. The flame propagation

is also slow so the pressure effects are small. When turbulence is generated, the

combustion speed increases and the pressure wave tends to pick up in magnitude

and become similar to a detonation. The profile of a blast wave from vapour

cloud deflagration is more likely an ”N” curve, as shown in 2.1b. The leading

wave is not a shock front, but a gradually increasing compression wave. Similarly,

peak overpressure po, positive duration td and specific impulse I are still used to

define the pressure profile. Moreover, the negative phase has a maximum of the

same order of the magnitude as that in the peak positive phase. The duration of

the negative phase is also of the same length as the positive phase.

2.1.2 Methods for explosion analysis

Predicting the blast damages to a piece of equipment and structure requires

firstly to calculate the time history of the blast wave produced in air. Nowadays.

computational advancement allow researchers to use sophisticated CFD tools to

examine the vapour cloud explosion and the blast wave characteristics. FLACSv9

and AutoReaGas are two CFD-based codes, developed by Century Dynamics

and Gexcon. Such codes usually comprise two solvers: a ”Gas explosion” solver

(Navier-Stokes) and a ”Blast” solver (Euler). The gas explosion solver is used for

analysis of the explosion process, including flame propagation, turbulence and the

effects of obstacles/congestions in the flow field. The blast solver is responsible for

accurate, efficient modelling of the shock phenomena and blast wave propagations.

The above two codes are not readily available at Imperial College. However,

recent updates of ? offered the Eulerian capabilities, which enables us to model

the blast wave without the explosion process. Modelling the blast wave is useful

in terms of wave propagation and most importantly the interaction with the

objects/structures. The Eulerian solver of ANSYS (AUTODYN) has also been

used in this project.

11

2.Literature review

In order to define a blast wave, in the absence of simulation of the vapour

cloud explosion process, there are two methods available: the TNT-equivalency

method and the TNO multi-energy method.

TNT-equivalency method

One way to predicting the blast wave from a vapour cloud explosion is to

estimate the mass of high explosive TNT that would produce the comparable

blast damage and similar blast wave time history. This is known as the TNT

equivalency method. Due to its simplicity and acceptable accuracy for most of

the vapour cloud explosions, it is still widely used today. However the method

should produce a reasonable prediction of the blast wave in the far-field, where the

blast wave from a vapour cloud explosion is transforming into an ideal blast wave

(detonation type). In the near - field the predicted blast wave can be significantly

different from an actual vapour cloud blast wave (Pritchard, 1989).

To estimate the equivalent TNT mass me from the energy of the vapour

release, a factor called TNT efficiency fe is used. It is the ratio, defined by the

total explosive heat of the equivalent TNT to the total combustion heat released

from the vapour cloud.

fe =HTNTme

Hvapourmvapour% (2.1)

In Equation 2.1, HTNT is the explosive heat of TNT, me is the equivalent

mass of TNT, Hvapour is the combustion heat of the fuel in the vapour cloud and

mvapour is the total mass of the vapour cloud. It was suggested by Pritchard

(Pritchard, 1989) that to estimate an explosive involving vapour cloud, the value

between 1% and 10% is usually used for TNT efficiency fe. The highest value

for deflagration is 10%, beyond this value it is representative for a detonation.

The major uncertainty of this method comes from value used for TNT efficiency.

Once it is determined, the equivalent mass of TNT me can be readily obtained

and the blast wave parameters at any distance from the explosion centre can be

found by lots published charts/tables or equations by Kinney and Graham (1985).

TNO multi-energy method

12

2.Literature review

(a) TNO overpressure chart (b) TNO duration chart

Figure 2.2: Blast waves parameters in TNO multi-energy method (Berg, 1985)

A more sophisticated approach was proposed by Berg (Berg, 1985). It is able

to predict the vapour cloud explosion blast wave with more realistic shape and

parameters.

This multi-energy method is based on the observation that the explosive

strength of vapour cloud primarily comes from the obstructed or partially con-

fined parts of the cloud. The blast from a vapour cloud is predicted by an idealised

explosion of which blast parameters are available in the form of charts shown in

Figure 2.2. Equations used to define the parameters in Figure 2.2 are presented

as follows (Berg, 1985):

13

2.Literature review

ΔP s =ΔPsP0

(2.2a)

t+ =t+c0

(E/P0)(1/3)(2.2b)

R =R

(E/P0)(1/3)(2.2c)

P0 = atmospheric pressure

c0 = atmospheric soundspeed

E = amount of combustion energy

R = stand− off distance

where ΔP s is the dimensionless maximum overpressure, t+ is the dimension-

less positive duration and R is the energy scaled distance to the explosion centre.

It is assumed that the strong blast only comes from the part of cloud that is

subjected to high degree of confinement and the rest of the cloud will just burn

out without making any significant contribution to the blast strength. Therefore

the blast from the confined and unconfined area covered by vapour cloud can be

modelled separately, using a high strength value for the partially confined area

and low strength for the unconfined area, and combine to produce the overall

results. Since the flame velocity in the unconfined part is relatively slow, and it

is very unlikely that two source of strong blasts would be initiated at the same

time, blasts generated from different part of the cloud could be somehow treated

independently (Pritchard, 1989).

The method assumes a semi-spherical vapour cloud with stoichiometric fuel

- air mixture and it has two characteristic parameters: the vapour cloud charge

energy E and the strength ΔPs. The charge energy E should be taken to be

equal to the total combustible heat of the fuel participating in the explosion and

the charge strength should be assumed to be maximum (Mercx et al., 2000).

In Figure 2.2, number 1 indicates a blast with insignificant effect and 10 for a

gas detonation. The shape will be changed according to the strength level as

depicted in Figure 2.2a. The major uncertainty comes from the estimation of

charge strength ΔPs, between 1 and 10. The estimation could be made based

14

2. Literature review

on the observed damage or experience. For an more sophisticated and accurate

approach, the correlation with experiments to quantify the charge strength can

be done according to the procedures laid out by Mercx (Mercx et al., 2000). The

correlation equation is:

ΔPs = a

[V BR · Lf

D

]bSl

2.7Sc0.7 (2.3)

In equation 2.3, V BR is the volume blockage ratio, which is the portion of the

volume occupied by the obstacles within the cloud. Lf is the flame path, which

is the longest distance from the point of ignition to an outer edge of the obstacle.

D is the average obstacle size (m). Sl is the laminar burning velocity (m/s). Sc

is a scale factor and a, b are correlation fitting constants. Equation 2.3 takes into

account of the effects from three influence factors: the boundary conditions, the

mixture reactivity and the explosion scale. Once the charge strength is deter-

mined, the selection of curves can be completed. The overpressure and positive

duration can be readily extracted from Figure 2.2 as a function of energy scaled

distance R.

2.2 Review of past incidents

2.2.1 Buncefield incident

On the morning of Sunday 11th December 2005, a vapour cloud explosion oc-

curred at the Buncefield Oil Storage Depot, Hemel Hempstead, Hertfordshire,

UK. The investigation immediately carried out by the Buncefield Major Incident

Investigation Board (BMIIB) identified that the explosion was a consequence of

the overflow of 250,000 litres of petrol from the storage tank 912 in the depot.

The cloud was approximately circular with a radius of 200 metres before the

explosion occurred, with average height of 2 metres (SCI, 2009).

The blast caused tremendous damage to the near and far field buildings. Due

to lack of any significant congestion in the site, the severity of the explosion had

not been anticipated in any major hazard assessment of the oil and gas storage

site before. The red line in Figure 2.3 represents the approximate boundary of

15


Figure 2.3: Buncefield site with location of buildings (Atkinson, 2011)

the vapour cloud immediately before explosion. The ignition point was found to

be the pump house.

Damage during the explosion is vast and involved lots of different objects.

Towards the edge of the vapour cloud, there are North Gate (NG) and Fuji

buildings which suffered severe damage, as shown in Figure 2.4a and 2.4b. In the

medium-far field a large warehouse structure (WH1) was located at the south of

the site, its damage is shown in Figure 2.4c after the claddings were removed.

Within the cloud, there are few buildings but lots of steel boxes (Figure 2.4d)

and drums were found scattered all over the site. The storage tanks (Figure

2.4e) were almost destroyed completely. Damage to the instruments within the

cloud suggested the overpressure was in excess of 2 bar. The overpressure rapidly

decreased outside of the cloud. This can be seen in Figure 2.4f, in which the car

furthest from the camera (i.e. nearest to the cloud) were clearly exposed to the

blast wave and crushed completely. However the car furthest from the cloud only

suffered slight damage at the rear.

During the investigation, both deflagration and detonation were compared

16


(a) Northgate Building (b) Fuji Building

(c) Warehouse - 1 (d) Steel Box

(e) Drum and Tanks (f) A line of cars

Figure 2.4: Buncefield Explosion Damages (SCI, 2009)

17


with the observed damages but neither is able to explain the explosion conclu-

sively. It was concluded in the joint industry project (SCI, 2009) and by Johnson

(Johnson, 2010) that due to the lack of congested pipe works within the vapour

cloud (which is a prerequisite for a strong vapour cloud explosion), the explo-

sion initially started as deflagration at the pump house. And when the flame

approached to the trees and undergrowths along the lanes around the site, the

combustion flame accelerated due to the turbulence generated by the wooden

branches, then developed in to detonation. At least part of the cloud has been

detonated afterwards.

Compared with the past major incidents, the 2005 Buncefield explosion of-

fered lots of evidence and damage for forensic studies and investigations. The

investigation work presented a great advancement in understanding vapour cloud

explosions. It also identified some areas worthy of fundamental researches.

2.2.2 Other major accidents

Buncefield is not the first vapour cloud explosion that occurred in the oil and gas

industries. A few previous incidents involving vapour cloud and gas explosions

are summarised below:

• Flixborough, UK, 1974 (Sadee et al., 1977), in which the failure of a sectionof 700 mm diameter temporary pipe released approximately 100 tonnes of

cyclohexane. The explosion caused significant damage to the building struc-

tures on site. The overpressure at the edge of the cloud was approximately

1 bar declining with distance as would be expected from a TNT charge

detonation. Within the cloud the pressure was estimated to be as high as

10 bar.

• Port Hudson, Missouri, US, 1970 (Burgess and Zabetakis, 1973), in whichthe pipeline failure released approximately 120,000 litres of propane. The

character and distribution of damage strongly resembled that at Buncefield

(SCI, 2009). Damages to buildings within the cloud are significant and

suggesting overpressure of at least 1 bar over the area covered by the cloud.

18


• Ufa, Ural Mountains, Russia, 1989 (Makhviladze and Yakush, 2002), inwhich the failure of a pipeline of 700 mm diameter released a total 2000-

10,000 tonnes of liquefied petroleum gas (LPG). Trains were damaged as if

they were clamped in a vice. Trees felled over a large area suggesting the

overpressure exceeded 1 bar.

• Naples, 1985 (Maremonti et al., 1999), in which approximately 700 tonnespetrol escaped from an over-filled tank, and a subsequent vapour cloud ex-

plosion occurred. Maximum pressure was indicated by the damaged objects

to be 0.48 bar. Windows were broken to a range of 1 km.

• Saint Herblain, France, 1991 (Lechaudel and Mouilleau, 1995), in which ahydrocarbon explosion occurred in a tank farm as a consequence of petrol

leakage. Explosion caused severe damages to the buildings in the surround-

ing area and also to the tanks on site. Overpressure indicators were analysed

and the maximum pressure of 0.25 bar was estimated.

• Texas City, Texas, U.S, 2005 (BP, 2005), in which a total of 28,700 litreshydrocarbons C5-C7 released into atmosphere and mixed with air. The

damage was focused in congested areas of the site where there was en-

hancement of turbulence promoting the combustion rate. The overpressure

was estimated to be over 1 bar within the congested area.

2.3 Blast wave and loading

Detonation and deflagration of a vapour cloud generate very different forms of

blast waves as shown in Figure 2.1a and 2.1b. The shape of the blast wave from

a deflagration will approach to detonation at the medium to far-field distance.

The blast wave from a detonation led by a shock front (with instantaneous rise

of pressure) is usually referred to as an ideal blast wave. Its parameters can

be defined by the TNT-equivalent method (with stand-off distance and charge

weight).

19


2.3.1 Blast wave characteristics

A detonation blast wave is usually referred to as an ideal blast wave or shock

wave. It is led by a shock front when travelling through the air and compresses

the fuel mixture or air in the front. The properties of the medium before and

after the shock front can be related by the Rankine-Hugoniot relations. The shock

wave characteristics can be defined by TNT explosion and scaling laws based on

the charge weight and stand-off distance.

2.3.1.1 Normal shock

x

x

x

x

TP

u

x

y

y

y

y

TP

u

y

upstream

Stationary Normal Shock

downstream

Figure 2.5: Stationary normal shock in a moving stream

The shock front of a blast wave is the determining factor in its characteristics.

The normal shock has extremely thin thickness and it increases the pressure P ,

temperature T and density ρ of the flow passing through instantly. As a result

the flow is decelerated rapidly across it. It is convenient to use the shock front

itself as the reference datum to study its behaviours as shown in Figure 2.5.

The Rankine - Hugoniot relationships

20


In a moving stream, flow temperature, density, pressure and velocity are re-

quired to describe the flow completely. Four equations 2.4a to 2.4d can be used to

define above properties. The direction of the flow can be considered by Equation

2.4e (White, 2003).

mass conservation : ρxux = ρyuy (2.4a)

linear momentum : Px − Py = ρyuy2 − ρxux2 (2.4b)energy : cpTx +

1

2ux

2 = cpTy +1

2uy

2 (2.4c)

perfect gas :PxρxTx

=PyρyTy

(2.4d)

entropy consideration :sy − sx

R=

k

k − 1 lnTyTx

− lnPyPx

(2.4e)

Mach number : Mx =ux√kRTx

& My =uy√kRTy

(2.4f)

In Equation 2.4, x and y are subscripts for denoting up and downstream

(or before and after shock front respectively). ρ is the density of the flow. P

is pressure and T is temperature of the flow stream. cp is the specific heat at

constant pressure and k is the ratio of specific hear constants. s is the entropy

in the flow stream and R is the universal gas constant. M is the Mach number

of the flow stream.

Assuming the properties of flow in the upstream (x) are known and the flow

system is adiabatic, Rankine-Hugoniot relationships can be derived from Equa-

tion 2.4 to relate the properties on both sides of the normal shock, thus the flow

conditions after the shock front can be determined. The entropy consideration

Equation 2.4e implies that the pressure in the downstream must be higher than

the upstream (Py/Px > 1), otherwise the change of entropy across the normal

shock will be negative and this violates the second law of thermodynamics. And

as a result the Mach number before the normal shock (in the upstream) must be

supersonic (Mx > 1) and the flow after the shock is subsonic (My < 1) (White,

2003).

21


PyPx

=2kMx

2 − (k − 1)k + 1

=7Mx

2 − 16

(2.5)

TyTx

=[2 + (k − 1)Mx2][2kMx2 − (k − 1)]

(k + 1)2Mx3

=(5 +Mx

2)(7Mx2 − 1)

36Mx2 (2.6)

uyux

=2 + (k − 1)Mx2(k + 1)Mx

2 =1 + 5/Mx

2

6(2.7)

Relationships giving the properties changes across the normal shock can be

derived from the above equations (Kinney and Graham, 1985). Equations 2.5,

2.6 and 2.7 define the ratios of pressure, temperature and flow velocity between

the up- and down-streams in terms of independent variable Mach number in the

upstream (Mx).

Explosive shock

atmx

atmx

atmx

x

TTPP

u

x

y

y

y

yxp

TP

uuu

yMoving Normal Shock

Medium after shock

Undisturbed Medium before shock

Figure 2.6: Explosive shock front moving in still air

The analysis of the stationary normal shock also applies to a moving shock

22


front generated in an explosion if we reverse the situation in which now the shock

front is moving through the still air at velocity ux and the undisturbed medium

ahead of the moving shock is at atmospheric condition. The downstream flow

then appears to follow the shock front at a lower speed up which is known as

the particle velocity. This defines the blast wind after passage of the shock wave

(Kinney and Graham, 1985).

p = Py − Px = 2k(Mx2 − 1)

k + 1Px =

7(Mx2 − 1)6

Px (2.8)

The peak overpressure p in the explosive shock is the difference of the pressures

across the shock front as shown in Equation 2.8. It can also be presented in terms

Mach number Mx and pressure Px in the upstream.

up = uy − ux (2.9a)upax

=2(Mx

2 − 1)(k + 1)Mx

=5(Mx

2 − 1)6Mx

2 (2.9b)(upax

)2=

(2/k)(Py/Px − 1)2(k + 1)(Py/Px) + (k + 1)

=25(Py/Px − 1)242(Py/Px) + 7

(2.9c)

Particle velocity (up) in the blast wave is the difference between the flow ve-

locities across the shock front as depicted in Equation 2.9a (Kinney and Graham,

1985). up/ax , the ratio of the particle velocity (up) to the local speed of sound

(ax) in the upstream (in the medium before shock wave) can be presented either

in terms of upstream Mach number Mx or ratio of absolute pressures Py/Px, as

shown in Equations 2.9b and 2.9c.

Pstag = Py

[1 +

(k − 1)(up/ax)22(Ty/Tx)

]k/(k−1)= Py

[1 +

(up/ax)2

5(Ty/Tx)

]3.5(2.10a)

pstag = Pstag − Px (2.10b)

Stagnation pressure (Pstag) is used in a flow to measure the sum of free stream

dynamic and static pressures. In a explosive shock it applies to the blast wind

23


following the shock front, and defined by Equation 2.10a (Kinney and Graham,

1985). The stagnation overpressure (pstag) which is the stagnation pressure minus

initial pressure is defined in 2.10b.

Reflected shock

When blast waves encounter a rigid surface of a object they will reflect from

it and, dependent on the size of the surface , diffract around it. The simplest case

is that the blast wave impinges on an infinitely large rigid surface at zero angle

of incidence β as shown in Figure 2.7.

x

x

x

TP

y

y

y

TP

Incident Shock

Reflected Shock

pu

xu

Rigid surface

rP

y

y

y

TP

ru

pu

Figure 2.7: Normal reflected shock on a rigid surface

In a normal reflection, the reflected pressure (Pr) can be defined in terms of

the pressures across the incident shock front Py and Px, as shown in Equation

2.11.

PrPx

=(Pr/Px)[(3k − 1)(Py/Px)− (k − 1)]

(k − 1)(Py/Px) + (k + 1) =Py(8Py − Px)Px(Py + 6Px)

(2.11)

The magnitude of a reflection shock can be also expressed in terms of reflection

coefficient Λ and for a normal reflection Λn, as defined in Equation 2.12.

24


Λn =prp

=Pr − PxPy − Px =

(3k − 1)(Py/Px) + (k + 1)(k − 1)(Py/Px) + (k + 1) =

8Py + 6PxPy + 6Px

(2.12)

where pr is the reflected overpressure. Equation 2.12 for the normal reflection

coefficient allows two limiting values to be determined. For a very weak shock

wave in which Py ≈ Px, the coefficient Λn is approaching to 2. While for a strongshock in which Py � Px, the coefficient approaches to 8. Thus the theoreticallimits for Λn can be found to be between 2 and 8. However, because of gas

dissociation effect at very close range (Smith and Hetherington, 1994), reflection

coefficients higher than 8 can be observed in actual blast. It can also be seen in

Figure 2.8 that for a very high incident overpressure the reflection coefficient can

be well above 8. The derivation of the expressions shown above can be found in

more details in “Explosive shocks in air” (Kinney and Graham, 1985) and it is

assumed that the specific heat ratio k = 1.4 for air (ideal gas).

0 10 20 30 40 50 60 70 80 90Angle of incident β

0

2

4

6

8

10

12

Reflection

coeffi

cientΛ

0.0689 bar

0.345 bar

2.068 bar

6.895 bar

20.685 bar

27.58 bar

34.475 bar

344.75 bar

Figure 2.8: Reflected pressure coefficients vs. angle of incidence (β) (USDOD,2008)

25


Equations 2.11 and 2.12 are only applicable when the blast wave impinges

perpendicularly to the surface, which is a normal reflection. The angle of incident

is zero (β = 0o) and the magnitude of reflection is maximum. When the reflecting

surface is parallel to the direction of the blast wave (β = 90o) the surface will

experience the minimum reflected pressure which is actually the incident pressure.

The magnitude of the reflection depends on both the incident peak pressure

and the angle of incidence. Typical reflection factors are depicted in Figure 2.8

(FEMA).

2.3.1.2 Blast wave and scaling law

A blast wave is generated when the surrounding medium (mostly air) is forcibly

pushed outwards by the high temperature and pressure gases in the vapour cloud

explosion. A detonation blast wave with the profile shown in Figure 2.9a is identi-

cal to a high explosive explosion thus it can be analysed by the TNT-equivalency

method. The shape of the deflagration blast wave from a VCE is shown in Figure

2.9b. In the far field the deflagration shape will gradually approach to a det-

onation when the wave propagates away from the centre. This is because the

high pressure portion of the deflagration wave is travelling at a higher local speed

of sound than the other parts of the blast wave with lower pressures, hence the

peak part will gradually catch up the leading compressive position and eventually

form a detonation blast wave. The peak overpressure of a blast wave will reduce

over the distance and its duration will be elongated due to the different velocities

throughout the blast wave.

A detonation blast wave is also known as an ideal blast wave. Its character-

istics are well understood and can be predicted based on the TNT-equivalency

method and the scaled distance Z defined in Equation 2.14. As mentioned be-

fore, four parameters are used to define an ideal blast wave: peak overpressure

po, arrival time ta, positive phase duration td and the specific impulse I. The

blast wave is leading by a normal shock front with peak overpressure po, and the

exponential decay of the overpressure can be described by Equation 2.13, finally

it will drop to zero at the end of the duration td. α is the shape factor which

determines the rate of the exponential decay, usually α = 1. For simplicity an

26


0.0 0.2 0.4 0.6 0.8 1.0

Time-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Overpressure

po

td

ta

Incident overpressure p

Equivalent form

(a) Detonation

0.0 0.2 0.4 0.6 0.8 1.0

Time-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Overpressure

po

td

tr

Incident overpressure p

(b) Deflagration

Figure 2.9: Shape of blast waves

27


alternative (equivalent) form can be taken in which the decay is assumed to be

linear and the impulse is preserved to be the same to the exponential form.

p = po(1− t

td

)�−αt/td (2.13)

In the above equation, t is time and � is the base of natural logarithm. α

is used to define the form of the shape of the blast wave. Many methods have

been used to obtain the characteristics of blast waves from explosions. I

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